The upper (lower) sign in (REF ) corresponds to a Neumann (Dirichlet) b.c. for \(\tilde{\varphi }\)  at the bang. This is closely related to the b.c. at the boundary of anti de Sitter (AdS) spacetime [1]}, [2]}: there too, if one conformally maps the spacetime to a portion of the infinite static (Einstein) cylinder, one finds a simple reflecting (Neumann or Dirichlet) b.c. for the field \(\tilde{\varphi }\) . In fact, the AdS b.c.s were originally derived [1]} by mapping AdS to a portion of the static Einstein cyclinder, and then noting that it could be analytically extended to form a two-sheeted spacetime, with a natural \(\mathbb {Z}_{2}\)  symmetry relating the sheets. Of course, this closely parallels our derivation, but with some noteable differences: in the AdS case, one analytically continues in the radial direction, and the resulting b.c.s make the AdS boundary an ordinary mirror (a timelike surface which reflects radially); while in our case, one analytically continues in the time direction, and the resulting b.c.s make the bang an exotic mirror (a spacelike surface which reflects in the time direction).
